Electrodynamics as a Theory of Persistent Stochastic Processes

Abstract: A process-theoretic approach to electrodynamics based on persistent Kac-type stochastic processes is developed. Finite-velocity stochastic propagation is taken as primary, while relativistic wave equations arise as emergent descriptions after analytic continuation of Telegrapher-type dynamics. The Dirac and Maxwell equations are interpreted as arising from closely related persistent propagation structures differing only in spin representation. The framework is not intended to modify the successful empirical predictions of quantum electrodynamics, but to provide a different underlying ontology. Particles and fields are not treated as primitive entities with fixed intrinsic properties. Instead, relativistic particle and field structures emerge as stable collective modes of coupled persistent stochastic dynamics. Mass and charge acquire interpretations respectively as persistence and stochastic coupling scales. Stationary bound states are interpreted as metastable persistent stochastic modes with nontrivial internal sector dynamics. Spontaneous emission is viewed as stochastic destabilization of such modes, while stimulated emission arises through resonant synchronization of persistent transition currents by incident radiation. Gauge interactions are introduced at the level of propagation-sector amplitudes prior to the emergence of observable probabilities. Radiative effects, including the anomalous magnetic moment of the electron, are interpreted as effective stochastic dressing of coupled matter--radiation processes. Some comments on gauge symmetry, equilibration and the Standard Model are also included.

• The paper introduces a process-theoretic framework where persistent Kac-type stochastic processes give rise to emergent Dirac and Maxwell equations.
• It reinterprets fundamental quantum parameters, positioning mass and charge as emergent properties tied to persistence and sector switching rather than intrinsic attributes.
• The framework clarifies radiative corrections and gauge symmetry by modeling quantum transitions as finite, coupled stochastic processes, offering a fresh take on quantum field theory.

Electrodynamics Reformulated via Persistent Stochastic Dynamics

Process-Theoretic Reinterpretation of Relativistic Quantum Theory

The paper proposes a process-theoretic reconstruction of electrodynamics and quantum mechanics, departing from canonical particle-field ontology. The central thesis is that persistent stochastic propagation, specifically Kac-type processes with finite velocity and sector switching, serves as the underlying basis for both Dirac and Maxwell equations. Relativistic wave equations are emergent phenomena, obtained via analytic continuation from the stochastic Telegrapher equation dynamics. Spin and mass are reinterpreted as structural and persistence parameters of these stochastic processes rather than innate attributes.

Foundations in Stochastic Mechanics and Kac Processes

The framework builds initially on Nelson's stochastic mechanics, where the quantum wavefunction encodes statistics of conservative Brownian motion. This non-relativistic formalism is generalized to the Kac process, where stochastic switching between propagation sectors occurs with finite speed $c$ and Poisson-distributed reversal rate $\lambda$.

After analytic continuation ($\lambda \rightarrow \frac{i m c^2}{\hbar}$), the Kac process yields dynamics corresponding to the Dirac equation. Notably, mass emerges as a stochastic parameter tied to persistence timescale rather than a fundamental entity (2605.24494). The process naturally incorporates multi-sector probability dynamics, aligning more closely with quantum field theory structures than Schrödinger-type mechanics.

Emergence of Relativistic Wave Equations and Sector Dynamics

Both Dirac and Maxwell equations are shown to arise as emergent descriptions of persistent stochastic propagation; for Dirac, the underlying process is governed by a spin-$\frac{1}{2}$ sector structure, while for Maxwell, the Riemann–Silberstein formulation recasts Maxwell’s equations in a Dirac-like, spin-$1$ framework [Bialynicki1996]. The vacuum photon dynamics are formulated as persistent stochastic propagation within helicity sectors, where the electromagnetic field becomes a spin-$1$ stochastic process.

Sector coupling (and conservation of total probability, but not individual sector probabilities) provides a dynamical analogy to particle creation, annihilation, and sector mixing found in field-theoretic models. Gauge interactions are conceptualized as stochastic couplings between process amplitudes, preceding the emergence of physical probabilities.

Interpretation of Quantum Parameters: Mass, Charge, and Radiative Structure

Mass and Charge as Emergent Stochastic Parameters

The process-theoretic framework reinterprets physical parameters:

• Mass: Persistence scale of sector switching in the Kac process; the Dirac mass term is thus an emergent property.
• Charge: Effective coupling strength between stochastic matter and radiation processes. This shifts the meaning from a primary, intrinsic attribute to a parameter regulating stochastic response and sector interaction.

Gauge principles are realized as invariance properties operating before the emergence of observable probability amplitudes.

Stationary States, Spontaneous and Stimulated Emission

Contrary to the static picture of stationary states in canonical quantum mechanics, even stationary solutions exhibit internal sector dynamics and osmotic motion. Excited states are interpreted as metastable persistent stochastic modes, with spontaneous emission mapped to stochastic instability and stimulated emission to resonant synchronization of transition currents by external fields. This mechanism provides a robust dynamical foundation for quantum transitions and their emission spectra.

Radiative Corrections and Anomalous Magnetic Moment

Radiative effects, including the electron's anomalous magnetic moment, are attributed to effective stochastic dressing from matter–radiation interaction. Rather than arising from divergent self-energy corrections of field-theoretic point particles, these corrections are finite emergent coefficients in the coupled stochastic dynamics. The leading term $a_e = \frac{\alpha}{2\pi}$ is identified as the weak-coupling approximation of stochastic spin response. This approach obviates parameter renormalization and shifts the focus toward finite emergent response properties.

Symmetry Structure, Gauge Invariance, and Standard Model Implications

Within this process-theoretic paradigm, symmetry principles are interpreted as emergent properties of internal sector representation structures. The spin and multiplet configurations in the Standard Model correspond to different persistent stochastic dynamics. Gauge symmetry, including possible non-Abelian components, may arise from largescale invariance of coupled sector dynamics, rather than being an imposed principle. Mass acquisition mechanisms (e.g., Higgs sector) are hypothesized to reflect persistent stochastic structural features, potentially allowing reinterpretation of spontaneous symmetry breaking as emergent from underlying sector persistence.

Practical and Theoretical Implications

This approach provides a concrete stochastic basis for relativistic quantum phenomena, with implications for foundational quantum ontology, renormalization, and the interpretation of quantum field theory. The formalism suggests observational equivalence with quantum field theory but introduces a radically distinct underlying ontology. Practical advances may be gained in understanding quantum processes, modeling radiative effects, and constructing new stochastic simulations for relativistic quantum systems.

Future directions include formal development of coupled matter–radiation persistent stochastic equations, exploration of non-Abelian gauge emergence, and modeling symmetry-breaking phenomena within this framework.

Conclusion

The process-theoretic ontology proposed yields structural and observational equivalence with established quantum field theories, providing a stochastic foundation for the emergence of relativistic wave dynamics, mass, and charge. Radiative and symmetry effects are reframed as finite emergent responses of persistent sector-level stochastic processes. This reformulation invites deeper investigation into quantum theory’s foundations, potentially offering new insight into the ontology and dynamics underlying the Standard Model and quantum electrodynamics.